Animated Solution for Physics - Rotational Motion: Consider a situation in which a ring, a solid cylinder and a solid sphere roll down on the same inclined plane without slipping. Assume that they start rolling from rest and having identical diameter. The correct statement for this situation.

Imagine you are standing at the top of a steep hill, holding three objects: a ring, a solid cylinder, and a solid sphere. They all have the exact same mass and the exact same radius. You release them at the exact same time. They roll down the hill without slipping. Which one reaches the bottom first? Which one is moving the fastest when it crosses the finish line?

This is a classic physics race, and the winner isn't determined by weight or size, but by how their mass is distributed! Let's dive into the beautiful physics of rolling motion to find out who takes the gold medal.

To solve this, we need to look at the energy of the system. At the top of the incline, all three bodies are at rest. They possess only gravitational potential energy, given by $mgh$, where $h$ is the height of the incline.

As they roll down, this potential energy is converted into kinetic energy. But here is the catch: because they are rolling without slipping, they aren't just moving forward (translating); they are also spinning (rotating). Therefore, the total kinetic energy at the bottom is the sum of translational kinetic energy and rotational kinetic energy.

Here, $v$ is the linear velocity of the center of mass, $I$ is the moment of inertia, and $\omega$ is the angular velocity.

We can simplify this equation to make it easier to compare the three bodies. First, we can express the moment of inertia $I$ in terms of the radius of gyration $k$, such that $I=m{k}^2$. The radius of gyration is simply a measure of how far the mass is distributed from the center.

Second, because the bodies are rolling without slipping, the linear velocity and angular velocity are perfectly locked together by the relation $v=\omega R$, or $\omega =\frac{v}{R}$.

Substituting these into our energy equation:

Notice how the mass $m$ cancels out completely! This means a heavy sphere and a light sphere will roll down at the exact same speed. Factoring out the velocity $v$, we get:

Solving for the final velocity $v$, we arrive at our master equation:

This equation tells us a profound truth: the final velocity depends entirely on the factor $\frac{k^2}{R^2}$. The smaller this factor, the smaller the denominator, and the larger the final velocity!

Now, let's calculate this factor for each of our competitors.

1. The Ring: A ring has all its mass concentrated at its outer edge. Its moment of inertia is $I=m{R}^2$. Therefore, $k^2=R^2$, and the factor $\frac{k^2}{R^2}=1$.

2. The Solid Cylinder: A solid cylinder has its mass spread evenly throughout its volume. Its moment of inertia is $I=\frac{1}{2}m{R}^2$. Therefore, the factor $\frac{k^2}{R^2}=\frac{1}{2}$.

3. The Solid Sphere: A solid sphere has its mass concentrated closest to its center compared to the other two. Its moment of inertia is $I=\frac{2}{5}m{R}^2$. Therefore, the factor $\frac{k^2}{R^2}=\frac{2}{5}$.

Comparing the results, we see that $v_{\text{sph}}>v_{\text{cyl}}>v_{\text{ring}}$.

The solid sphere wins the race! Because its mass is concentrated closer to the center, it has the smallest moment of inertia. It requires less energy to spin, leaving more of the initial potential energy to be converted into pure, forward speed. The ring, on the other hand, is the hardest to spin, so it ends up being the slowest.